Download 49, 482 Chapter 8: Techniques of Integration 48. 49, 29, Evaluate

49,
482
Chapter 8: Techniques of Integration
Mean temperature
Evaluate the integrals in Exercises 29-32 (a) without using a trigonometric
substitution, (b) using a trigonometric substitution.
29, y d y
x^x
^
/'
tdt
Vl6 - j/2 x Jx
31
estimate the annual mean air temperature in Fairbanks, Alaska. The
National Weather Service's official figure, a numerical average of the
daily normal mean air temperatures for the year, is 25.7°F, which is
slightly higher than the average value of /(x).
V4 + x2
30,
32.
vV ■
Evaluate the integrals in Exercises
9 - x2
I dx
x(9 - x2) dx
36. V9~
dx
,
Heat capacity of a gas Heat capacity C v is the amount of heat required to
raise the temperature of a given mass of gas with constant volume by 1°C,
measured in units of cal/deg-mol (calories per degree gram molecular
weight). The heat capacity of oxygen depends on its temperature T and
satisfies the formula
50.
33-36.
34. x d x
33
C v = 8.27 + 10~5 ( 2 6 T - 1.87r2).
Use Simpson's Rule to find the average value of C v and the temperature at
which it is attained for 20° < T < 675°C.
Trig or. egrals
35
Evaluate the integrals in Exercises 37-44.
311
y
,
37.
J
40.
Use Simpson's Rule to approximate the average
value of the temperature function 2TT
(x - 101) + 25
f ( x ) = 37 sin
365 for a 365-day year. This is one way to
sin x d x
cos5 x
Fuel efficiency An automobile computer gives a digital readout of fuel
consumption in gallons per hour. During a trip, a passenger recorded the
fuel consumption every 5 min for a full hour of travel.
51,
tan x sec x d x
sin3 x cos4 x d x 39. / tan4 x
sec2 x d x
Time
sin 5 0 cos 6 0 d O
41.
42.
cos 30 cos 3 0 d O
'Vt~<
43.
y
VT+ cos U/2) <ft
44.
+ 1<#
y(
Gal/h
Time
Gal/h
0
2.5
35
2.5
5
10
15
20
25
30
2.4
2.3
2.4
2.4
2.5
2.6
40
45
50
55
60
2.4
2.3
2.4
2.4
2.3
45. According to the error-bound formula for Simpson's Rule, how many
subintervals should you use to be sure of estimating the value of
~ dx
In 3
36 ft
by Simpson 's Rule with an error of no more than 10~4 in absolute value?
(Remember that for Simpson's Rule, the number of subintervals has to be
even.)
46.
A brief calculation shows that if 0 < x < 1, then the second derivative of
f ( x ) = 4- x4 lies between 0 and 8. Based on this, about how many
subdivisions would you need to estimate the integral of / from 0 to 1 with
an error no greater than 10~3 in absolute value using the Trapezoidal Rule?
A direct calculation shows that
47,
2 sin2x d x = rr.
a. Use the Trapezoidal Rule to approximate the total fuel consumption during the hour.
b. If the automobile covered 60 mi in the hour, what was its fuel
efficiency (in miles per gallon) for that portion of the trip?
52. A new parking lot To meet the demand for parking, your town has
allocated the area shown here. As the town engineer, you have been
asked by the town council to find out if the lot can be built for
$11,000. The cost to clear the land will be $0.10 a square foot, and the
lot will cost $2.00 a square foot to pave. Use Simpson's Rule to find
out if the job can be done for $ 11,000.
Oft
o
48.
49,
54 ft
How close do you come to this value by using the Trapezoidal Rule with n
= 6? Simpson's Rule with n = 6? Try them and find out.
You are planning to use Simpson's Rule to estimate the value of the
integral
51 ft
49.5 ft
Vertical spacing = 15 ft
54 ft
n
f(x) dx
with an error magnitude less than 10~5. You have determined that |/(4)(x)| <
3 throughout the interval of integration. How many subintervals should
you use to assure the required accuracy? (Remember that for Simpson's
Rule the number has to be even.)
48.
64.4 ft
67.5 ft
42 ft
Ignored